How to extrude from hexagon but keep edge lines straight
you can use cursor for this
1-put it in center of hexagon (select 2 points and shift+S cursor to selected) and change transform pivot point to cursor
2-hit E to extrude then press S scale in towards the center(extrusion does transformation too so by pressing s or clicking, it just make the edge without moving it)
3-again use cursor and rotate 180 degrees along y axis that face you just made
4-GG to edge slide and adjust how far you want to go along red line you mentioned